Interesting Topological Spaces in Algebraic Geometry

D. Zack Garza


Space, but Which One?

  • You run into a “space” in the wild. Which one is it?
  • How many possible spaces could it be?
  • How much information is needed to specify our space uniquely?

A Space in the Wild

One Motivation: Physics

Does your space have genus?
Does your space have curvature or singularities?

Another Application: Data

Can you only measure low-dimensional slices/projections?
  • Possible to fit data to a high-dimensional manifold
  • Makes clustering/grouping easier
    • (Here, slice with a hyperplane)
Does your space abstractly parameterize something else?
  • Extract info about an entire family of objects and how they vary.
  • Also useful for outlier detection!

Where to Start

  • What structure does your space have?

  • What can you “measure” locally?

  • How might it vary in ways you can’t measure?

  • Important question before attempting to classify:

    1. What does “space” mean?
    • Need to pick a category to work in.
    1. What does “which” mean?
    • How to distinguish? Need an equivalence relation!

Q1: Types of Spaces

  • Plan: compare classification theorem in topology and algebraic geometry
  • Hopefully see some fun spaces along the way!
  • But first: address classification and the notion of “sameness”

The Greeks: Conics

  • Early classification efforts: conic sections.

    • Apollonius, 190 BC, Ancient Greeks
  • Key idea: realize as intersection loci in bigger space

    • (Projectivize \(\mathbb{R}^2\).)

  • A conic is specified by 6 parameters:

\[\begin{align*} A x^{2}+B x y+C y^{2}+D x+E y+F=0 \\ \\ Q :=\left(\begin{array}{ccc} A & B / 2 & D / 2 \\ B / 2 & C & E / 2 \\ D / 2 & E / 2 & F \end{array}\right), \quad \mathbf{x} = [x, y, 1] \\ \implies \mathbf{x}^t Q \mathbf{x} = 0 \end{align*}\]

\(\det(Q)\) Conic
\(<0\) Hyperbola
\(=0\) Parabola
\(>0\) Ellipse
  • Each conic is a variety

    • Can obtain every conic by "modulating* 6 parameters.
  • \(\mathbb{R}^6\): too much information: scaling by a nonzero \(\lambda \in \mathbb{R}\) yields the same conic, so reduce the space \[ [A, B, C, D, E, F]\in \mathbb{R}^6 \mapsto [A: B: C: D: E: F] \in \mathbb{RP}^5 .\]

  • Important point: \(\mathbb{RP}^5\) is a projective variety and a smooth manifold! Tools available:

    • Dimension (what does a generic point look like?)
    • Tangent and cotangent spaces, differential forms
    • Measures, metrics, volumes, integrals
    • Intersection theory (Bezout’s Theorem!), subvarieties, curves
    • Linear algebra and Combinatorics (enumerative questions)
  • We can imagine a moduli space of conics that parameterizes these:
Moduli Space of Conics


Some Calc III review:

General form: \[\begin{array}{l}\scriptsize A x^{2}+B y^{2}+C z^{2}+2 F y z+2 G z x+2 H x y+2 P x+2 Q y+2 R z +D=0 \\ \\ \text{Setting}\,\, E:= \left[\begin{array}{llll} A & H & Q & P \\ H & B & F & Q \\ G & F & C & R \\ P & Q & R & D \end{array}\right]\,\, e:= \left[\begin{array}{lll} A & H & G \\ H & B & F \\ G & F & C \end{array}\right] \\ \Delta := \operatorname{det}(E) \end{array}\]

(discriminants), the equation becomes \(\mathbf x^t E \mathbf x = 0\) and we have a classification:

Classification of quadrics

What is the moduli space? It sits inside \(\mathbb{R}^{10}\), possibly \(\mathbb{RP}^{9}\) but not in the literature!


  • Problem: infinitely many points in these moduli spaces correspond to the same “class” of conic
Partition a Moduli Space
  • How to address: Klein’s Erlangen program, understand the geometry of a space by understanding its structure-preserving automorphisms.

  • For topological spaces: a Lie group acting on the space.

  • Can then “mod out” by the appropriate morphisms to (hopefully) get finitely many equivalence classes

What Does “Space” Mean?

Some Setup

  • Algebraic Variety: Irreducible ,zero locus of some family \(f\in \mathbb{k}[x_1, \cdots, x_n]\) in \(\mathbb{A}^n/\mathbb{k}\).
    • Equivalently, a locally ringed space \((X, \mathcal{O}_X)\) where \(\mathcal{O}_X\) is a sheaf of finite rational maps to \(\mathbb{k}\).
  • Projective Variety: Irreducible zero locus of some family \(f_n \subset \mathbb{k}[x_0, \cdots, x_n]\) in \(\mathbb{P}^n/\mathbb{k}\)
    • Admits an embedding into \(\mathbb{P}^\infty/\mathbb{k}\) as a closed subvariety.
  • Dimension of a variety: the \(n\) appearing above.
  • Topological Manifold: Hausdorff, 2nd Countable, topological space, locally homeomorphic to \(\mathbb{R}^n\).
    • Equivalently, a locally ringed space where \(\mathcal{O}_X\) is a sheaf of continuous maps to \(\mathbb{R}^n\).
  • Smooth Manifold: Topological manifold with a smooth structure (maximal smooth atlas) with \(C^\infty\) transition functions.
    • Equivalently, a locally ringed space where \(\mathcal{O}_X\) is a sheaf of smooth maps to \(\mathbb{R}^n\).
  • Algebraic Manifold: A manifold that is also a variety, i.e. cut out by polynomial equations. Example: \(S^n\).
  • Manifolds will be compact and without boundary, varieties are (probably) smooth, separated, of finite type.

Impossible Goal

  • Pick a category, understand all of the objects (identifying a moduli “space”) and all of the maps.
    • Understand all topological spaces up to ???
      • Homeomorphism?
      • Diffeomorphism?
      • Homotopy-Equivalence?
      • Cobordism?
    • Understand all algebraic and/or projective varieties up to
      • Biregular maps?
      • Birational maps?
      • Locally ringed morphisms?

Classification in Topology

Topological Category

  • Classifying manifolds up to homeomorphism: stratify “moduli space” of topological manifolds by dimension.

  • Dimensions 0,1,2,3:

    • Smooth = Top. See smooth classification.
  • Dimension 4:
    • Topologically classified by surgery, but barely, and not smoothly.
Surgery in Action
  • Dimension \(n\geq 5\):
    • Uniformly “classified” by surgery, s-cobordism, with a caveat:
    • \(\pi_1\) can be any finitely presented group – word problem
    • Instead, breaks homotopy type of a fixed manifold up into homeomorphism classes
Surgery Classification

Smooth Category

Generally expect things to split into more classes.

  • Dimension 0: The point (terminal object)
  • Dimension 1: \(\mathbb{S}^1, \mathbb{R}^1\)
  • Dimension 2: \(\left\langle\mathbb{S}^2, \mathbb{T}^2, \mathbb{RP}^2 \mid \mathbb{S}^2 = 0,\,\,3\mathbb{RP}^2 = \mathbb{RP}^2 + \mathbb{T}^2 \right\rangle\).
    • Classified by \(\pi_1\) (orientability and “genus”). Riemann, Poincaré, Klein.

  • Dimension 2: closed + orientable \(\implies\) complex
    • Uniformization: Holomorphically equivalent to a quotient of one of three spaces/geometries:
      • \(\mathbb{CP}^1\), positive curvature (spherical)
      • \(\mathbb{C}\), zero curvature (flat, Euclidean)
      • \(\mathbb{H}\) (equiv. \(\mathbb{D}^\circ\)), negative curvature (hyperbolic)
    • Stratified by genus
      • Genus 0: Only \(\mathbb{CP}^1\)
      • Genus 1: All of the form \(\mathbb{C}/\Lambda\), with a distinguished point \([0]\), i.e. an elliptic curve.
        • Has a topological group structure!
      • Genus \(\geq 2\): Complicated?

Doesn’t capture holomorphy type completely.

3-manifolds: Thurston’s Geometrization

  • Geometric structure: a diffeo \(M\cong \tilde M/\Gamma\) where \(\Gamma\) is a discrete Lie group acting freely/transitively on \(X\) (as in Erlangen program)
  • Decompose into pieces with one of 8 geometries:
    • Spherical \(\sim S^3\)
    • Euclidean \(\sim \mathbb{R}^3\)
    • Hyperbolic \(\sim \mathbb{H}^3\)
    • \(S^2\times \mathbb{R}\)
    • \(\mathbb{H}^2\times \mathbb{R}\)
    • \(\widetilde{\mathrm{SL}(2, \mathbb{R})}\)
    • “Nil”
    • “Sol”
  • Proved by Perelman 2003, Ricci flow with surgery.
Ricci Flow with Surgery
More Ricci Flow
  • 4-manifolds: classified in the topological category by surgery, but not in the smooth category
    • Hard! Will examine special cases of Calabi-Yau
    • Open part of Poincaré Conjecture.
  • Dimension \(\geq 5\): surgery theory, strong relation between diffeomorphic and s-cobordant.

Toward Algebraic Manifolds: Berger’s Classification

  • Every smooth manifold admits a Riemannian metric, so consider Riemannian manifolds

  • Here \(H\leq \mathrm{SO}(n)\) is the holonomy group:

  • Berger’s classification for smooth Riemannian manifolds, one of 7 possibilities. \[ \tiny \begin{array}{|c|c|c|c|c|} \hline n=\operatorname{dim} M & H & \text { Parallel tensors } & \text { Name } & \text { Curvature } \\ \hline n & \mathrm{SO}(n) & g, \mu & \text {orientable} & \\ \hline 2 m(m \geq 2) & \mathrm{U}(m) & g, \omega & \textbf{Kähler} & \\ \hline 2 m(m \geq 2) & \mathrm{SU}(m) & g, \omega, \Omega & \textbf{Calabi-Yau} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & \mathrm{Sp}(m) & g, \omega_{1}, \omega_{2}, \omega_{3}, J_{1}, J_{2}, J_{3} & \textbf{hyper-Kähler} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & (\mathrm{Sp}(m) \times \mathrm{Sp}(1)) / \mathbb{Z}_{2} & g, \Upsilon & \text {quaternionic-Kähler} & \text {Einstein} \\ \hline 7 & \mathrm{G}_{2} & g, \varphi, \psi & \mathrm{G}_{2} & \text {Ricci-flat} \\ \hline 8 & \operatorname{Spin}(7) & g, \Phi & \operatorname{Spin}(7) & \text {Ricci-flat} \\ \hline \end{array} \]

Types in bold: amenable to Algebraic Geometry. \(G2\) shows up in Physics!

  • Ricci-flat, i.e. Ricci curvature tensor vanishes
    • (Measures deviation of volumes of “geodesic balls” from Euclidean balls of the same radius)

Classification in Algebraic Geometry

Enriques-Kodaira Classification

Work over \(\mathbb{C}\) for simplicity, take all dimensions over \(\mathbb{C}\).

  • Minimal model program: classifying complex projective varieties.

  • Stratify the “moduli space” of varieties by \(\mathbb{k}-\)dimension.

  • Dimension 1:

    • Smooth Algebraic curves = compact Riemann surfaces, classified by genus
    • Roughly known by Riemann: moduli space of smooth projective curves \(\mathcal{M}_g\) is a connected open subset of a projective variety of dimension \(3g-3\).

We’ll come back to these!

  • Dimension 2:
    • Smooth Algebraic Surfaces: Hard. See Enriques classification.

    • Setting of classical theorem: always 27 lines on a cubic surface!

    • Example Clebsch surface, satisfies the system \[ \begin{array}{l} x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0 \\ \\ x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0 \end{array} \]

Clebsch Surface with 27 Lines

Interesting Space: Elliptic Curves

  • Equivalently, Riemann surfaces with one marked point.
  • Equivalently, \(\mathbb{C}/\Lambda\) a lattice, where homothetic lattices (multiplication by \(\lambda \in \mathbb{C}- \{0\}\)) are equivalent.
  • Generalize to \(\mathbb{C}^n/\Lambda\) to obtain abelian varieties.
Elliptic Curves as Plane Curves
Complex Curves as Real Surfaces

Interesting Space: Moduli of Elliptic Curves

  • \(\mathcal{M}_g\): the moduli space of compact Riemann surfaces (curves) of genus \(g\), i.e. elliptic curves.
Approximately the Actual Space

Dimension 2: Algebraic Surfaces

Generalize to Moduli Space of Surfaces?
  • Definition: Kodaira Dimension

    • \(X\) has some canonical sheaf \(\omega_X\), you can take some sheaf cohomology and get a sequence of integers (plurigenera)

\[\begin{align*} P_{\mathbf{n}} (X) &:= h^0(X, \omega_X^{\otimes \mathbf{n}}) \quad n\in \mathbb{Z}^{\geq 0} \\ \\ \implies \kappa(X) &:= \limsup_{\mathbf n \to \infty} {\log P_{\mathbf n}(X) \over \log(\mathbf{n}) } \end{align*}\]

Dimension 2: Algebraic Surfaces

Every such surface has a minimal model of one of 10 types:

\(\kappa = -\infty\) (2 main types)

    1. Rational: \(\cong \mathbb{CP}^2\)
    1. Ruled: \(\cong X\) for \(\mathbb{CP}^1 \to X \to C\) a bundle over a curve.
    • Called “ruled” because every point is on some \(\mathbb{CP}^1\).
    1. Type VII
Our Old Friend the Hyperboloid

\(\kappa = 0\) (Elliptic-ish, 4 types)

    1. Enriques (all (quasi)-elliptic fibrations)
    1. Hyperelliptic
    • Taking Albanese embedding (generalizes Jacobian for curves) yields an elliptic fibration
      • (i.e. a surface bundle, potentially with singular fibers)
    1. \(K3\) (Kummer-Kahler-Kodaira) surfaces
    1. Toric and Abelian Surfaces:
    • 2 dimensional abelian varieties (projective algebraic variety + algebraic group structure).
    • Compare to 1 dimensional case: all 1d complex torii are algebraic varieties,
      • Riemann discovered that most 2d torii are not.
    1. Kodaira Surfaces

\(\kappa = 1\): Other elliptic surfaces

  1. Properly quasi-elliptic.

Elliptic fibration, but almost all fibers have a node.

\(\kappa = 2\) (Max possible, “everything else”)

  1. General type

Interesting Space: Toric Varieties

  • Definitions:
    • Define a complex torus as \(\mathbb{T} = (\mathbb{C}^{\times})^n \subseteq \mathbb{C}^n\)

    • Can be written as the zero set of some \(f\in \mathbb{C}[x_0, \cdots, x_n]\) in \(\mathbb{C}^{n+1}\).

      Generalizes to algebraic groups over a field: \((\mathbb{G}_m)^n\) (analogy: maximal torus/Cartan subalgebra in Lie theory)

    • Toric variety: \(X\) contains a dense Zariski-open torus \(\mathbb{T}\), where the action of \(\mathbb{T}\) on itself as a group extends to \(X\).

The Wide World of Polytopes
  • Flavor: spaces modeled on convex polyhedra
  • Examples: bundles over \(\mathbb{CP}^n\).
  • Why study:
    • Model spaces by rigid geometry, generalize things like Bezier curves
      • Some are determined by rigid combinatorial data (“fan”, or polytopes)
      • Combinatorial data for constructions in mirror symmetry, e.g. Calabi-Yaus (1/2 of one billion threefolds!)

Kahler Manifolds/Varieties

  • As complex manifolds:
    • A symplectic manifold \((X, \omega)\) with an integrable almost-complex structure \(J\) compatible with \(\omega\).
    • Yields an inner product on tangent vectors: \(g(u, v) := \omega(u, Jv)\) (i.e. a metric)

  • Examples: all smooth complex projective varieties
    • But not all complex manifolds (exception: Stein manifolds)
  • Specialize to Calabi-Yaus: compact, Ricci-flat, trivial canonical.
    • Calabi’s Conjecture and Yau’s field medal: existence of Ricci-flat Kahlers (Calabi-Yaus)

Trivial canonical \(\implies\) exists a nowhere vanishing top form = top wedge of \(T^* X\) is the trivial line bundle


  • Another from Berger’s classification, special case of Kahler
Calabi-Yau “Strings”
  • Applications: Physicists want to study \(G_2\) manifolds (an exceptional Lie group, automorphisms of octonions)
    • Part of \(M\)-theory uniting several superstring theories, but no smooth or complex structures.
    • Indirect approach: compactify an 11-dimension space, one small \(S^1\) dimension \(\to\) 10 dimensions
      • 4 spacetime and 6 “small” Calabi-Yau
    • Superstring theory: a bundle over spacetime with fibers equal to Calabi-Yaus.

Roughly, genera of fibers will correspond to families of observed particles.


  • As manifolds:
    • Ricci-flat: vacuum solutions to (analogs of) Einstein’s equations with zero cosmological constant

    • Setting for mirror symmetry: the symplectic geometry of a Calabi-Yau is “the same” as the complex geometry of its mirror.

    • Yau, Fields Medal 1982: There are Ricci flat but non-flat (nontrivial holonomy) projective complex manifolds of dimensions \(\geq 2\).

  • As varieties: the canonical bundle \(\Lambda^n T^* V\) is trivial

  • Compact classification for \(\mathbb{C}-\)dimension:

    • Dimension 1: 1 type, all elliptic curves (up to homeomorphism)
    • Dimension 2: 1 type, \(K3\) surfaces
  • Dimension 3: (threefolds) conjectured to be a bounded number, but unknown.
    • At least 473,800,776!
Example (from Jim Bryan): The Bananafold